Pollack: Palindromic Sums of Proper Divisors

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Paul Pollack, Palindromic Sums of Proper Divisors, Integers 15A (2015), article A13.

Overview

For a fixed base $g\ge2$, Pollack asks how often the proper-divisor sum $s(n)=\sigma(n)-n$ has a palindromic base-$g$ expansion. A number is $k$-nearly-palindromic when its first $k$ digits reverse its last $k$ digits, with numbers below $g^{2k}$ included by definition (§1, p. 1). Theorem 1 (p. 2) bounds the upper density of $n$ for which $s(n)$ is $k$-nearly-palindromic by $O_g(1/\log k)$. Since every palindrome is $k$-nearly-palindromic for every $k$, this proves that palindromic values of $s(n)$ occur on a density-zero set of inputs.

The proof (§2, pp. 2–7) combines distributional and digit arguments. Lemma 2 (pp. 2–3), using Shapiro’s cited theorem, gives continuous limiting distributions $D_{a,q}$ for $s(n)/n$ in each residue class modulo $q$. Lemma 3 (pp. 3–4) shows that $D_{a,q}$ depends on $a$ only through $\gcd(a,q)$; its moment argument uses Lemma 4 and the divisor expansion in (1). Lemma 5 (pp. 4–5), deduced from Watson’s cited estimate, says that each fixed $W$ divides $\sigma(n)$ for almost all $n$. Lemma 6 (p. 5) bounds Davenport’s distribution mass on an interval $I$ by $O(1/\log(2+|I|^{-1}))$. In the proof of Theorem 1, divisibility by $g^k$ makes the last $k$ digits $B$ of $s(n)$ determine $n\equiv-B\pmod{g^k}$; the reversed digits constrain $s(n)/n$ to an interval of length $O_g(kg^{-k})$ in (2) (p. 6). The progression count (3) and gcd restriction (4) (p. 6), followed by Lemmas 3 and 6 (p. 7), yield the stated bound.

Section 3 treats other arithmetic functions. Lemmas 7–8 (p. 8) give concentration and maximal-fiber estimates for $\omega$ and $\Omega$; the paper calls density zero for their palindromic values an easy consequence and sketches it, leaving the details to the reader. Theorem 9 (pp. 8–10) bounds the upper density of $n$ with $k$-nearly-palindromic $d(n)$ by $O_g(g^{-2k/3})$ when $g$ is not a power of $2$; the proof combines those estimates with equidistribution of multiples of $\log 2/\log g$. Proposition 10 (§3.2, p. 10), quoted from Pollack and Vandehey, concerns compositions of $\varphi,\sigma,\lambda$: the preimage of a thin set is thin. Corollary 11 (p. 11) applies it to palindromes, including numbers made palindromic by deleting trailing zeros. Section 4 (p. 11) recalls the general density-zero preimage assertion for $s$ as a conjecture of Erdős, Granville, Pomerance and Spiro [8, Conjecture 4], not as a result of this paper.

Relation to E955

This source bears on Problem 955.

In E955’s notation, take $A=P_g$, the positive integers palindromic in base $g$. The count $|P_g\cap[1,x]|\asymp_g x^{1/2}$ is noted in §1 (p. 1), and Theorem 1 proves that $s^{-1}(P_g)=\{n:s(n)\in P_g\}$ has density zero. The proof also handles this particular target through the larger sets of $k$-nearly-palindromic values.

The usable mechanism for E955 is the combination of $\sigma(n)\equiv0\pmod{g^k}$ for almost all $n$ (Lemma 5), progression distributions for $s(n)/n$ (Lemmas 2–3), and a small-interval mass bound (Lemma 6). It enters after a target’s structure links a residue of $s(n)$ to a narrow interval for $s(n)/n$, as palindrome reversal does in (2). An arbitrary density-zero $A$ need supply no such link; the paper gives no bound for $s^{-1}(A)$ based solely on $|A\cap[1,x]|=o(x)$. Proposition 10 applies to $\varphi,\sigma,\lambda$ and their compositions, not to $s$. Section 4 (p. 11) cites E955’s assertion as Conjecture 4 of Erdős, Granville, Pomerance and Spiro and says that nothing nontrivial toward it is known without structural assumptions on $A$.